Summary
We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.
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This work has been supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 “Stochastische Mathematische Modelle”